P UBLICATIONS MATHÉMATIQUES DE L ’I.H.É.S.
F ABIEN M OREL
V LADIMIR V OEVODSKY
A
1-homotopy theory of schemes
Publications mathématiques de l’I.H.É.S., tome 90 (1999), p. 45-143
<http://www.numdam.org/item?id=PMIHES_1999__90__45_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
http://www.numdam.org/
by FABIEN MOREL, VIADIMIR VOEVODSKY
CONTENTS
1. Preface . . . 45
2. Homotopy category of a site with interval . . . 45
2.1. Homotopy theory of simplicial sheaves . . . 46
2.2. A localization theorem for simplicial sheaves . . . 70
2.3. Homotopy category of a site with interval . . . 85
3. The A -homotopy category of schemes over a base . . . 94
3.1. Simplicial sheaves in the Nisnevich topology on smooth sites . . . 95
3.2. The A -homotopy categories . . . 105
3.3. Some realization functors . . . 119
4. Classifying spaces of algebraic groups . . . 122
4.1. Generalities . . . 122
4.2. Geometrical models for Bg^G in ^ (S) . . . 133
4.3. Examples . . . 137
1. Preface
In this paper we begin to develop a machinery which we call A1-homotopy theory of schemes. All our constructions are based on the intuitive feeling that if the category of algebraic varieties is in any way similar to the category of topological spaces then there should exist a homotopy theory of algebraic varieties where affine line plays the role of the unit interval. For a discussion of the main ideas on which our approach is based we refer the reader to [32].
2. Homotopy category of a site with interval
In this section we prove a number of general results about simplicial sheaves on sites which will be later applied to our study of the homotopy category of schemes.
In the first part (Section 1) we describe the main features of the homotopy theory of simplicial sheaves on a site. Many results of this part can be found in [20] and [17], [18]. Surprisingly, there are some nontrivial things to be proven in relation to basic functoriality of the homotopy categories of simplicial sheaves. This is done in Section 1.
In Section 2 we prove a general theorem which shows that there is a "good"
way to invert any set of morphisms in the simplicial homotopy category of a site. Here
"good55 means that the resulting localized category is again the homotopy category for some model category structure on the category of simplicial sheaves. The results of this
sections remain valid in a more general context of model categories satisfying suitable conditions of being "locally small" but we do not consider this generalizations here.
In Section 3 we apply this localization theorem to define a model category structure on the category of simplicial sheaves on a site with interval (see [31, 2.2]).
We show that this model category structure is always proper (in the sense of [2, Definition 1.2]) and give examples of how some known homotopy categories can be obtained using this construction.
All through this section we use freely the standard terminology associated with Quillen's theory of model categories. The notion of a model category which we use here first appeared in [9] and is a little stronger than the one originally proposed by Quillen.
To avoid confusion we recall it here.
Definition 0.1. — A category ^ equipped with three classes of morphisms respectively called weak equivalences, cofibrations and fibrations is called a model category if the following axioms hold :
• MC1 W has all small limits and colimits;
• MC2 Iff and g are two composabk morphisms and two of f, g or go f are weak equivalences, then so is the third;
• MC3 If the morphismfis retract of g and g is a weak-equivalence, cojibration orfibration then so is f;
• MC4 Any jibration has the right lifting property with respect to trivial cofibrations (cofibrations which are also weak equivalences) and any trivial fibration (afibration which is also a weak equivalence) has the right lifting property with respect to cofibrations;
• MC5 Any morphism f can be junctorialy (in f) factorised as a composition p o i where p is afibration and i a trivial cofibration and as a composition q o j where q is a trivial fibration and j a cofibration,
The only differences between these axioms and Quillen's axioms CM1, .... CM5 of a closed model category are the existence of all small limits and colimits in axiom MC1 instead of just finite limits and colimits, and the existence ofjunctorial factorisations in axiom MC5.
Recall that a site is a category with a Grothendieck topology, see [13, II. 1.1.5].
All the sites we consider in this paper are essentially small (equivalent to a small category) and, to simplify the exposition, we always assume they have enough points (see [13]).
2.1. Homotopy theory of simplicial sheaves
Simplicial sheaves
Let T be a site. Denote by Shv(T) the category of sheaves of sets on T. We shall usually use the same letter to denote an object of T and the associated sheaf
because in our applications the sites we shall consider will have the property that any representable presheaf is a sheaf, in which case the canonical functor T —^ Shv(T) is a fully faithfull embedding. Let A°^Az<T) be the category of simplicial objects in Shv(T) ; this category is a topos (^[13]) and in particular has all small limits and colimits and internal function objects (the latter means that for any simplicial sheaf J^ the functor
^ ^ ^ x ^ has a right adjoint ^ ^ Hom(^, ^ )).
An object ^ of A°^(T), i.e. a functor A°^ -> Shv{T) is determined by a collection of sheaves of sets J^^ n^O, together with morphisms
^ : ^ ^ ^ _ 1 7 2 ^ 1 Z = 0 , . . , 7 Z
^:jr^jr^ n^o z=o,..,/2
called the faces and degeneracies which satisfy the usual simplicial relations ([22]).
To any set E one may assign the corresponding constant sheaf on T which we also denote by E. This correspondence extends to a functor from the category A^Sets of simplicial sets to A^Shi^T). For any simplicial set K the corresponding constant simplicial sheaf is again denoted K.
The cosimplicial object A ^ A^6%z<T) a> o»
n ^ A72
defines as usual a structure of simplicial category on A^Shv^T) (see [26]) with the simplicial function object S(—, —) given by
S(^T, ^)=Hom^T)^ x A', ^).
Observe that for a simplicial sheaf JT and an object U o f T the simplicial set S(U, ^T) is just the simplicial set of sections of <^T over U.
For any simplicial sheaf J^ and any n ^ 0, let S^^ C ^ be the union of the images of all degeneracy morphisms from SK'\_^ to J^, i.e.
qy^deg _ , .n-\ n-\, ^ .
^ n - ^•=0^ (^ n-i)'
For any simplicial sheaf ^T and any n ^ 0, one defines its n-th skeleton skn(J^) C S^
as the image of the obvious morphism ^ x A" —^ S^. We extend this definition to the case n= - 1 by setting sk,^) :=0. For example, (sW\^ is equal to ^T^.
The skeleton functor ^ ^-> skn{^) has a right adjoint ^ ^ coskn(J^) which is called the yz-th coskeleton functor.
A simplicial sheaf ^ is said to be of simplicial dimension ^ n if ^ x A" —> 3^
is an epimorphism, or equivalendy if skn{^)=J^. We will identify sheaves of sets with simplicial sheaves of simplicial dimension zero.
For any n ^ 0, let <9A" be the boundary of the n-th standard simplicial simplex.
The following straightforward lemma (which can be proven using points of T and the corresponding lemmas for simplicial sets) provides the basis for skeleton induction and will be used in Section 3 below.
Lemma 1 . 1 . — For any monomorphism of simplicial sheaves f: ^ —» ^ denote by skn[f) the union off(^) and skn{^) in ^ . Then for any n^O the square
((^U^^)XA-)U(^^^^ —— ^-i(/)
n
y. x a" -^ *,(/) 1 1
is cocartesian.
The simplicial model category structure
Recall that a point of a site T is a functor x* : Shv(T) —> Sets which commutes with finite limits and all colimits.
Definition 1.2. — Let f\ SF —> ^ be a morphism of simplicial sheaves.
1. f is called a weak equivalence if for any point x of the site T the morphism of simplicial sets x"{f) : x"(Jy) —> x^(^} is a weak equivalence;
2.fis called a cqfibration if it is a monomorphism;
3. f is called a fibration if it has the right lifting property with respect to any cofibration which is a weak equivalence (see [26, 1.5] for the definition of the right- (or left-) lifting property).
Denote by W^ (resp. C, FJ the class of (simplicial) weak equivalences (resp. cofibration, (simplicial) fibrations).
Remark 1.3. — Let JST be a simplicial sheaf. One defines its n-th homotopy sheaf n^^T) as the sheaf of pointed sets over J%"o associated to the presheaf {XQ : U —> J%^) i—> 7^(J^T(U)5 XQ) (of course, it is a sheaf of groups (resp. abelian groups) over J^o for n > 1 (resp. n > 2)). A morphism of simplicial sheaves/: ^ —> ^ is a weak equivalence if and only if for any n ^ 0 the square
nw —> iW)
^o —— ^
i i
is cartesian. Using this fact one can see that / is a weak equivalence if and only if x^f is a weak equivalences for all x in a conservative set of points of T (see [13] for this notion).
Theorem 1.4. — For any small site T the triple (W^ C, F^) gives the category ^ShvfT) the structure of a model category.
Proof. — It was shown in [18, Corollary 2.7] that the triple (W,, C, F,) defines a closed model structure on A^^A^(T) in the sense of Quillen. The proof of existence of factorizations given in [18] shows that they are functorial and therefore the stronger axioms which we use are satisfied.
This model category structure is called the simplicial model category structure on A^Shv^T). In the sequel, if not otherwise stated, we shall always consider the category A^6%y(T) endowed with that model category structure. We shall sometimes use the terminology simplicial weak equivalence (resp. fibration, cofibration) if we want to insist that we use this model category structure.
We denote the corresponding homotopy category by ^^(T).
Remark 1.5. — The simplicial model category structure on A^»Sfo(T) is proper {cf[2, Definition 1.2]). This is proven in [19].
By the axiom MC5 of model categories, we know that it is possible to find a functor Ex : A^Shu^T) —> ^Shv^T) and a natural transformation Id —> Ex such that for any J^T the object Ex{J^) is fibrant and the morphism SF —> Ex(^) is a trivial cofibration.
Definition 1.6. — A resolution functor on a site T is a pair (Ex, 9) consisting of a functor Ex : A^Shv^T) —^ A^Az^T) and a natural transformation 6 : Id —> Ex such that for any
J^T the object Ex(J^) is fibrant and the morphism J%" —> Ex(^>K") is a trivial cofibration.
Remark 1.7. — It is not hard to check that the functor which sends a simplicial set to the corresponding constant simplicial sheaf preserves weak equivalences. It gives us for any T an "augmentation35 functor ^^Sets) —^ ^(T). Any point x of T gives a functor ;c*:<^(T) —> ^^Sets) which splits this augmentation functor.
If we consider the category of simplicial sheaves on T as a symmetric monoidal category with respect to the categorical product then it is a closed symmetric monoidal category ((/[21]) because of the existence of internal function objects. In more precise terms, for any pair of objects (^ , Ss ) 6 {A^Shv^T) )2 the contravariant functor on A^<T):
JT ^ Hom^sw^ X ^ ^ )
is representable by an object denoted by Hom(^ , §&), and called the internal function object from ^ to 3^. The following lemma says that, in the terminology of [16, B.3], the model category structure we consider on A^Shv^T) is an enriched model category structure:
Lemma 1.8.
1. For any pair (i: ^ —> ^ y j : SK" —> ^) ofcofibrations, the obvious morphism P(zj) : (^ x ^)IU xjr (^ x JT) ^ ^ x ^
zj ^ cofibration which is trivial if either i or j is.
2. For any pair of morphisms { i : JK" —> ^ ,p : S —> ^) such that i is a cofibration and p a jibration the obvious morphism
Hom(^ , <T) -^ Hom(^, ^) x^jy ,^) Hom(^ , J8) is a jibration which is trivial if either i or p is.
3. For any pair of morphisms { i : SK" —r ^,j& : S —> ^) such that i is a cofibration and p a jibration the obvious morphism of simplicial sets
S(^, ^) -^ S(^, ^) xs^ ,^) S(^, ^)
is a Kan jibration which is trivial if either i or p is.
Proof. — It is an easy exercise in adjointness to prove that 1) implies 2) and 3).
One proves 1) by reducing to the corresponding lemma in the category of simplicial sets using points of T.
Remark 1.9. — Lemma 1.8 clearly implies that the model category structure on A^Shv^T) is a simplicial model category structure : indeed, the third point in this lemma is precisely axiom SM7 of [26, 11.2].
Lemma 1.10. — Letf: SK' —r ^ he a morphism between jibrant simplicial sheaves. Then the following conditions are equivalent:
1. fis a simplicial homotopy equivalence (i.e. there exists g'. ^ —> J^ such that fog and go fare simplicially homotopic to identity);
2. f is a weak equivalence;
3. for any object U C T the map of (Kan) simplicial sets:
S(U,/): S(U, JT) ^ S(U, ^)
is a weak equivalence (in fact a homotopy equivalence).
Proof. — The implication (2) => (1) is standard: one factorizes first f as a trivial cofibration followed by a (trivial) fibration and applies [26, Cor. 11.2.5]. (1) => (3) follows easily from the canonical isomorphism S(U, Hom(^, J^)) ^ Hom(/^, 3^{U}\ To prove (3) =^ (2) we note from [13, IV6.8.2] that any point ^ o f T i s associated to a pro-object {Ua} of the category T. Then x*(f) is a filtering colimit of weak equivalences and thus a weak equivalence.
Local fibrations and resolution lemmas
Besides the classes of cofibrations, fibrations and weak equivalences there is another important class of morphisms F/,, in ^Shv{T) which is called the class of local fibrations.
Definition 1 . 1 1 . — A morphism of simplicial sheaves f : JT —> ^ is called a local fibration (resp. trivial local fibration) if/or any point xofTthe corresponding morphism of simplicial
sets x*{J^) —> x*{^) is a Kan fibration (resp. a Kan fibration and a weak equivalence).
A list of the most important properties of local fibrations can be found in [17].
We will only recall the following result. For simplicial sheaves ^, ^ denote by 7C(^T, ^) the quotient of Hom{J^, ^)=So(^T, ^) with respect to the equivalence relation generated by simplicial homotopies, i.e. the set of connected components of the simplicial function object S(^, ^\ and call it the set of simplicial homotopy classes of morphisms from JT to ^f. One easily checks that the simplicial homotopy relation is compatible with composition and thus one gets a category nA^Shv^T) with objects the simplicial sheaves and morphisms the simplicial homotopy classes of morphisms. For any simplicial sheaf JT denote by nTriv/J^ the category whose objects are the trivial local fibrations to J^ and whose morphisms are the obvious commutative triangles in n^Shv(T). From [6, §2] this category is filtering.
Lemma 1.12. — For any simplicial sheaf ^, the category nTriv/J^ is essentially small, i.e. equivalent to a small one.
Proof. — Let's say that a simplicial sheaf ^ is (T, ^-bounded if for each n ^ 0 and each U € T the cardinal of the set ^(U) is less than or equal to that of &^veT,meN#^^(V). The full subcategory of (T, ^-bounded simplicial sheaves is clearly essentially small. Thus to prove the lemma it suffices to prove that for any trivial local fibration/: (^! -> ^ there is a (T, ^-bounded simplicial sheaf ^ ' and a morphism g : (^ ' —> ^ such that/o^r is a trivial local fibration. This fact is proven as follows. Let n ^ 1 and S C ^ a sub-simplicial sheaf which is (T, ^-bounded and such that for each i G {0,..., n - 1} the morphism of sheaves:
S. , -^ Hom{9A\ §& )o x^^^ ^
is an epimorphism (observe that & —> ^ is a trivial local fibration exactly when one has this property for any i ^ 0). Now there is an (T, ^-bounded subsheaf S, C ^, whose image by the morphism
^ ^ Hom(Q^\ ^)o x^,^^ ^
is Hom(9/^1, c^ )o X^W^A" j^} ^ n' t^lls foll0^ easily from the fact that the latter sheaf is (T, ^-bounded (observe it is a subsheaf of (^ n-\f ^ ^n)' Gal1 ^ ' the
sub-simplicial sheaf of ^ generated by Ss and S^. It is clear that Ss ' is (T, J^T)- bounded and has the same property as Sy up to i = n. By induction we get the result.
Proposition 1.13. — For any simplicial sheaves J%^ ^, with ^ locally fibrant, the canonical map:
colim^^^^rriv/^^(^', ^) -^ Hom^y^ , ^) is a bijection.
For the proof see [65 §2] for sheaves on topological spaces and [18, p. 55] in the general case.
Remark 1.14. — One of the corollaries of Proposition 1.13 is the fact that for any pair (X,Y) of sheaves of simplicial dimension zero the map
Homsh^(X, Y) -^ 7^m^(T)(X, Y)
is bijective. In other words, the obvious functor Shv(T) —> ,^^(T) is a full embedding.
An important class of local fibrations can be obtained as follows. Let^: X —> Y be a morphism of sheaves of sets. Denote by G {f) the simplicial sheaf such that
C(A=X^
and faces and degeneracy morphisms are given by partial projections and diagonals respectively. Then / factors through an obvious morphism C (/) —> Y which we denote pf.
Lemma 1.15. — The morphism pf is a local fibration. It is a trivial local jibration if and only iffis an epimorphism.
Proof. — Since T has enough points, it is sufficient to prove the lemma for T the category of sets in which case it is obvious.
The following two "resolution lemmas55 will be used below to replace simplicial sheaves by weakly equivalent simplicial sheaves of a given type.
Lemma 1.16. — Let y be a set of objects in Shv(T) such that for any U in T there exists an epimorphism F —>• U with F being a sum of elements in y. Then there exists a functor
<&^ : A^<Ste(T) —> A°^»Sfo(T) and a natural transformation <S>y —> Id such that for any ^ one has
1. for any n^O the sheaf of sets ^y(^\ is a direct sum of sheaves in y 2. the morphism 0^(^<) —> ^ is a trivial local fibration.
Proof. — For a morphism f'.^—^^f define 0^(/) by the cocartesian square U F X ( 9 A " ——> jy
i i
U F x A - —— 0^(/)
where the coproduct is taken over the set of all commutative squares of the form F x <9A" ——> ^
F x A" ——> ^
with n ^ 0 and F in y. Let 0^(/) be the canonical morphism 0^(/) —^ ^<.
Set Ol^C/) to be 0^(0^(/)) and let O^^/) be the corresponding morphism
^y (/) ~^ ^ ' ^e get a sequence of simplicial sheaves 0^(/) and monomor- phisms 0^(/) —^ ^y[f) and we set 0^(/) to be the colimit of this sequence.
This construction gives a functorial decomposition of any morphism / of the form
jr-.<&^(/)^^.
One verifies easily that the functor ^ ^-> 0^(0 —> ^f) satisfies the conditions of the lemma by using the fact ([13, IV6.8] that any point x of T is associated to a pro-object {FoJ with each Foe G y.
Remark 1.17. — Lemma 1.16 applied to the class of representable sheaves shows, using Lemma 1.1, that the smallest full subcategory of ^/T) which contains all representable sheaves and which is closed under isomorphisms, homotopy cofiber and direct sums is ^(T) itself.
Lemma 1.18. — Let ^ be a simplicial sheafand ?Q : J^o ~^ ^o ^ an epimorphism of sheaves. Then there exists a trivial local fibration p : JT' —> 3^ such that po is the ^ero component ofP^
Proof. — Consider ?Q as a morphism of simplicial sheaves J^o —> S^. Then construct its decomposition in the same way as in the proof of Lemma 1.16 using the inclusions U X 9^ —^ U x A" with U running through all objects of T and n being strictly greater than zero.
Homotopy limits and colimits
Let S^ be a small category. For any functor ^ : S7 -» A^Shu^T) we may define by the usual formulas (cf [3, XI.4.5, XII.3.7]) its homotopy limit and its homotopy colimit which gives us functors
holim^ : ^Shv^f -. ^Shv{T) hocolim^ : ^Shv^ -^ A^Shv{T) where holim^J^ is the sheaf of the form
U^Ao&m^(^(U))
and hocolim^jy is the sheaf associated with the presheaf of the form U ^ hocolim^{^^S}\
Lemma 1.19. — For any junctor J^ : S7 —> ^Shv(T) and any simplicial sheaf ^, there is a canonical isomorphism
Hom{hocolim^ , ^) ^ holim^pHom^ , ^\
and in particular therms a canonical isomorphism of simplicial sets
^(hocolim^, ^) ^ holim^S{^, ^).
Similarly, there are canonical isomorphisms
Hom{^f , holim^) ^ holim^Hom(^f , J^), and
S(^ , holim^) ^ holim^^ , ^T).
Lemma 1.20. — For any functor ^ : ^ —^ A^Sh^T) and any point x of T, the simplicial set x'(hocolim^^} is canonically isomorphic to the simplicial set hocolim^{S^\ If ^7 is a finite category the same holds for holim^.
Corollary 1.21. — Let ^, ^f be junctors ^ -> ^Shv^T) andfa natural transformation
^T-> ^. Then:
1. if for any i G ^ the morphismf(i) is a cofibration (resp. a weak equivalence) then the morphism hocolim^{f) is cofibration (resp. a weak equivalence);
2. if^ is a right filtering category (cf[3, XII.3.5P, then the obvious morphism:
hocolim^S^ —> colim^S^
is a weak equivalence.
Proof. — The first point and the third one are easy corollaries of Lemma 1.20 and [3, XII, 3.5, 4.2, 5.2]. The second point is an easy exercise in adjointness using Lemmas 1.19, 1.10 and [3, XI, 5.5, 5.6].
Proposition 1.22. — Let ^, ^ be junctors £7 —> A^Shi^T) and f a natural transformation J%" —> ^ such that all the simplicial sheaves ^(i), ^ {i) are pointwise jibrant and the morphisms f(i} are fibrations. Then holim{f) is afibration. In particular if all the sheaves J^T(z) are fibrant then holim^jy is jibrant.
Proof. — Follows from [22, XI, 5.5, 5.6], Lemma 1.8(3) and the obvious fact that 9\—, holim^—) = holim^y\—, —).
Unlike the theory of homotopy colimits the theory of homotopy limits for simplicial sheaves on sites is different from the corresponding theory for simplicial sets because the analog of Lemma 1.20 does not hold for infinite homotopy limits.
As a result holim functor may not preserve weak equivalences even between systems of pointwise fibrant objects unless the objects are actually fibrant. An example of such a situation for an infinite product is given below. A more sophisticated example is given in 1.30.
Example 1.23. — Let T be a site with precanonical topology i.e. such that any representable presheaf is a sheaf. Assume that there exists a family of coverings pi : Vi —> pt of the final object of T such that for any U in T the intersection of images of Hom(U, Ui) in pt=Hom(U,pf) is empty (such a family can be found for example in the site associated with any profinite group which is not finite). Consider the simplicial sheaves ^\=G(U^ —> pt) (see definition prior to Lemma 1.15) and let Ex be a resolution functor on ^Shv^T). We claim that the canonical morphism n^\ —> Yl^^i ls not a weak equivalence. Indeed, by Lemma 1.15 each ofJ%"/s is weakly equivalent to the final object and therefore Y[ExJ^^ is weakly equivalent to the final object as well. On the other hand our condition on U^s implies that the product ]~[J%T is empty.
Eilenberg-MacLane sheaves and Postnikov towers
In this section we give a reformulation of the main results of [22, Ch. V] for the case of simplicial sheaves. In this context there are two noticeable differences between simplicial sheaves and simplicial sets. The first is that the weak homotopy type of a simplicial sheaf can not be recovered from the weak homotopy type of its Postnikoff tower unless some finitness assumptions are used (Example 1.30). The second is that a simplicial abelian group object is not necessarily weakly equivalent to the product of Eilenberg-MacLane objects corresponding to its homotopy groups (Theorem 1.34).
We adopt the following convention concerning complexes with values in an abelian category ^: a chain complex G^ is one whose differential has degree —1 and a cochain complex C* is whose differential has degree +1. If G^ is a chain complex, we shall denote G* its associated cochain complex with C^ '.=0^.
For a sheaf of simplicial abelian groups ^ on T denote by n^) the presheaf of the form U i-^ 7C,(^(U), 0). Similarly, for a chain complex of sheaves of abelian groups C^ denote by H.(C^) the presheaf U ^ H,(G^(U)).
Let N(^) be the chain complex of sheaves of abelian groups on T obtained from a simplicial abelian group ^ by applying the functor of the normalized complex (see [22, p. 93]) pointwise. Then we have ^(^)=H^.(N(^)). The functor N has a right adjoint F ([22, p. 95]) and we get the following result ([22, Th. 22.4]).
Proposition 1.24. — (N^ F) is a pair of mutually inverse equivalences between the category of complexes of sheaves of abelian groups A with A, = 0 for i < 0 and the category of sheaves of simplicial abelian groups.
Remark 1.25. — For a complex A which does not satisfy the condition A,=0 for i < 0 the composition N o r maps A to the truncation of A of the form N o r(A), = A, for i > 0, N o r(A)o = ker{do : AQ -> A_i) and N o r(A), = 0 for i < 0.
One defines the Eilenberg-MacLane objects associated with a sheaf of abelian groups A as K(A, ri) = r(A[n]) where A[n] is the chain complex of sheaves with the only nontrivial term being A in dimension n.
Denote the category of chain complexes of sheaves of abelian groups on T by Compl(AbShv^}. Recall that a morphism of cochain complexes f : C^ —> C^
is called a quasi-isomorphism if the corresponding morphisms of homology sheaves aH^C'J —» ^H.(G^) are isomorphisms for all i G Z. The localization of the category Compl{AbShv{T)) with respect to quasi-isomorphisms is called the derived category of chain complexes of sheaves on T and denoted by D(AbShv(T)).
For any chain complex of sheaves G^ let nTriv/C^ be the category whose objects are epimorphisms of complexes C^ —^ C* which are quasi-isomorphisms and whose morphisms are the obvious homotopy commutative triangles of complexes. The same method as the one used in the proof of Lemma 1.12 shows that nTriv/C is a (left) filtering category, essentially small. This implies that the derived category D{AbShv(T)) obtained from Comp{AbShv(T)) by inverting all the quasi-isomorphisms is indeed a category, in which the set of morphisms from C^ to D^ is given by the colimit:
^^C^WT^/C^0^ D*)
where 7c(—, —) denotes the set of homotopy classes of morphisms of chain complexes.
Recall that the hypercohomology H*(U, C*) of an object U o f T with coefficients in a cochain complex of sheaves G* is the graded group of morphisms Hom{Z, C^)
in the derived category of (chain) complexes of sheaves on T. The following almost tautological result provides an interpretation of hypercohomology groups in terms of simplicial sheaves (for a proof see [6, §3 Theorem 2]).
Proposition 1.26. — Let G* be a cochain complex of sheaves of abelian groups on T. Then for any integer n and any object U of T one has a canonical isomorphism IT(U, G*) =Hom^ ^(U, r(C*[7z])). In particular if G* =A is a sheaf of abelian groups we have IP(U, A) =Hom^ ^, K(A, n)\
For a simplicial sheaf ^ denote by P^(^) the simplicial sheaf associated with the presheaf U ^ (^T(U))^ where K ^ K^=7m(K -^ ^(K)) is the functor on simplicial sets defined in [22, p. 32]. The following result is a direct corollary of [22, 8.2, 8.4].
Proposition 1.27. — Let ^ be a locally fibrant simplicial sheaf. Then the sheaves P^^
are locally jibrant and the morphisms
^ ^ p(^
P^W -^ p(^
are local fibrations.
Iff: jy —> ^ is a weak equivalence of locally jibrant simplicial sheaves then for any n^O the morphism P^/) is a weak equivalence.
Remark 1.28. — Let ^ be a pointwise fibrant simplicial sheaf i.e. a simplicial sheaf such that for any U in T the simplicial set ^(U) is a Kan complex. Then the simplicial sheaf P^^T is pointwise fibrant. For any U and T and a point x € ^(U) one has
7C,(P^(U), x) = TC^(U), x) for i < n 7C,(P(W(U), x) = colim^ ^u^(^(U), ^) -^ ^W^ ), ^) for z = n 7l,<P(W(U)^)=0 f o r z > 7 2 where the colimit in the middle row is taken over all coverings W = {Vj —> U} of U and 7l,W^ ) , x ) = n^<^(U/), x\
Definition 1.29. — The tower of local fibrations (P^, P^1)^ -^ P^^) ^o^W
^ <? locally jibrant simplicial sheaf S^ is called the Postnikov tower ofJ^.
Functors P^ do not take fibrant simplicial sheaves to fibrant simplicial sheaves.
As a result of this fact the homotopy limit holim^Ex^'^T) of the tower of fibrant objects associated to the Postnikov tower of ^ is not in general weakly equivalent to J^ as shown in the following example.
Example 1.30. — Let T be the site of finite G-sets where G = Y[^Z/2 is the product of infinitely many copies of Z/2. Consider the constant simplicial sheaf JT on T which corresponds to the product of Eilenberg-MacLane spaces of the form n>o^(^/2? i) (it is also the product of the corresponding Eilenberg-MacLane sheaves in the category of sheaves). Then P^JT is weakly equivalent to rL^>oK(Z/2, i) and one can easily see that for any resolution functor Ex the homotopy limit holimn^Ex^^) is weakly equivalent to ^= Y[^Ex{K(Z/2, z)). We claim that the sheaf associated to the presheaf U i-> 7Co(^(U)) is nontrivial while the corresponding sheaf for ^ is clearly trivial. By Proposition 1.26 we have for any U in T
nWV))='[[Hl(U,Z/2)
i>0
Let T be the generator of H1 (Z/2, Z/2) and p,: : G -> Z/2 the projection to the i-th multiple. Consider the element a= FlA*^) m ^oG^O^)). This element does not become zero on any covering of the point and therefore gives a nontrivial element in the sections of the sheaf associated to U »—>- 7lo(^%"(U)).
Definition 1.31. — A site T is called a site of finite type if for any simplicial sheaf ^ on T the canonical morphism J%" —> holim^QEx^P^jy) is a weak equivalence.
Our next goal is to show that any site satisfying a fairly weak finiteness condition on cohomological dimension is a site of finite type in the sense of Definition 1.31. In order to do it we will need a description of simplicial sheaves with only one nontrivial
"homotopy group55 which is also of independent interest.
Definition 1.32. — Let ^ be a simplicial sheaf We say that SK" has only nontrivial homotopy in dimension d^ 0 if the following condition holds:
1. for any U in T, any x € ^T(U) and any n ^ 0, n \d the sheaf of sets on T/U associated with the presheaf'V/U i-» 7C^(^(V), x) is isomorphic to the point.
We say that ^ has only one nontrivial abelian homotopy group in dimension d ^ 1 if it has only nontrivial homotopy in dimension d and for any U in T and any x € ^(U) the sheaf of groups on T/U associated with the presheaf V/V i—^ TC^(^(V), x) is abelian (this condition is of course only meaningful for d=\).
The forgetful functor from the category of sheaves of simplicial abelian groups on T to the category of simplicial sheaves (of sets) on T has a left adjoint which we call the functor of free abelian group and denote by Z : ^Shv^T) —^ A^AbShv^T). For any simplicial sheaf J^ the sheaf of simplicial abelian groups Z(J^) is the sheaf associated with the presheaf U i—^ Z(^(U)) where Z(^T(U)) is the free abelian group generated by the simplicial set ^(U) (in [22] the functor Z is denoted by C : K ^ C(K)).
Proposition 1.33. — Let J^ be a simplicial sheaf which has only one nontrivial abelian homotopy group in dimension d^ 1. Denote by ^(JK") the fiber product
^W ——> P^(Z(^))
pt ——> Z
Then the obvious morphism J^T —» 3T(J%r) is a weak equivalence.
Proof. — For any point x of T one has ^*(P^(Z(Jr)))=(C(^*^))" (where the right hand side is written in the notations of [22, Def. 8.1]) which shows that it is enough to prove the proposition in the case of simplicial sets. For any simplicial set K the homotopy groups of the simplicial abelian group C(K) are the homology groups of K and by our assumption on JT, Hurewicz Theorems ([22, Th. §13]) and the main property of functors K^K1 ([22, Th. 8.4]) we conclude that (G^JT))" ^ x^ x Z which implies the statement of the proposition.
For jy satisfying the conditions of Definition 1.32 (2) we define a sheaf CT^T) as the sheaf associated with the presheaf U i—» H^(<^(U);Z). Using Hurewicz Theorems ([22, Th. §13]) one can verify immediately that for any U in T such that ^(U) is not empty and any x G J%"(U)o there is a canonical isomorphism between <m^(J%")T/u ^d the sheaf on T/U associated with the presheaf V ^—> 71;^%" (V), x).
The simplicial sheaf ^^(J^)) has a canonical structure of a sheaf of simplicial abelian groups, the morphism P^(Z(J^r)) —> Z is a surjective homomorphism and its kernel is canonically weakly equivalent to r(<m^(J%r) \d\). Thus the complex of sheaves N(P^(Z(J^T))) has two nontrivial homology groups namely dH. = Z and aH. =CT^(J^T).
Therefore it defines a morphism in the derived category of complexes of sheaves on T of the form Z —> aj^(J^)[d+ 1] and the projection P^(Z(^)) —^ Z splits if and only if this morphism is zero. Combining these observations we get the following result.
Theorem 1.34. — Let J%T be a simplicial sheaf whose only nontrivial homotopy group OK^(Jy) is abelian and lies in dimension d ^ 1. Any such J%" defines a cohomology class r\jy C H^fr^ (%(J%T)) and the pair (a7C^(<^)_, T\jy) determines J%" up to a weak equivalence.
If in addition J%" is fibrant then
( 0 if the restriction of V[jy to U is not zero 7io(^(U))=<[^^^ otherwise
Corollary 1.35. — Let j3T be a fibrant simplicial sheaf satisfying the conditions of Definition 1.32 for some d > 1 and let U be an object ofT such that for any sheaf of abelian groups F on T and any m^d one has H"(U, F) =0. Then jto(^(U)) =pt.
Sheaves with only one nontrivial homotopy group are related to Postnikov towers as follows.
Proposition 1.36. — Let ^ be a locally fibrant simplicial sheaf and p : ^ —> ^ be a local fibration weakly equivalent to the local fibration P^JT —^ P^-1^. Then for any U in T and any point ^ in S^ (U)o the fiber J^ of p over ^ considered as a sheaf on T/U has only one nontrivial homotopy group in dimension d (which is abelian if d ^ 2).
Proof. — Follows by the use of points from [22, Cor. 8.7].
Theorem 1.37. — Let T be a site and suppose that there exists a family (A^o of classes of objects ofT such that the following conditions hold:
1. Any object U in A^ has cohomological dimension ^ d i.e. for any sheaf T on T/U and any m> d one has H^, F) =0.
2. For any object V ofT there exists an integer dy such that any covering ofV in T has a refinement of the form {U^ —> V} with U^ being in A^..
Then T is a site of finite type.
Proof. — Let 3^ be a simplicial sheaf on T. Denote by p^ : GP^^) -^
GP^"1^^) a tower of fibrations weakly equivalent to the tower of local fibrations P^jy —> V^~^^. This tower is then pointwise weakly equivalent to the tower (Ex^J^)) for any resolution functor Ex on simplicial sheaves and since homotopy limits preserve pointwise weak equivalences of Kan simplicial sets and homotopy limit of a tower of fibration is weakly equivalent to the ordinary limit we conclude that to prove the theorem we have to show that the canonical morphism S^ —> lim^ GP^^T is a weak equivalence. We may further assume that SK" is a fibrant simplicial sheaf.
It is easy to see that our claim will follow if we show that the sheaves CT()(^) and CTo(lim^Q GP^^) are isomorphic for all J^ (to deduce the same fact for TC, one then replaces JT by the simplicial sheaf of pointed maps from any model of the i-sphere to J%").
By the second condition of the theorem any object in T has a covering consisting of objects in A^ for some d. Therefore it is sufficient to verify that for any d ^ 0 and any U C A^ the canonical map CTo(J^)(U) —> CTo(lim^ GP^^)^ is an isomorphism.
By definition of P^ for any i ^ 0 we have
CTo(P^) = CTo(GP^) = CTo(^T)
which immediately implies that the map in question is a monomorphism. The fact that it is an epimorphism follows from the standard criterion for a map of presheaves to give an epimorphism of sheaves, Lemma 1.38 below and the exact form of condition (2) of the theorem.
Lemma 1.38. — Let U be an object in A^. Then 7io(lim GP^J^U)) -^ 7Co(GP^jr(U))
z>0
^ (272 isomorphism.
Proof. — Let j^ : K^ -^ K^-1), z ^ 1 be a sequence of Kan fibrations of Kan simplicial sets and d be such that for any m ^ d and any x € K^ one has
^(O^)-1^))^. Then the map 7Co(lim.K^) -^ Tio(K^) is bijective. Combining this fact with Corollary 1.35 and Proposition 1.36 we get the statement of the lemma.
Remark 1.39. — We do not know of any example of a site where each object has a finite cohomological dimension but condition (2) of Theorem 1.37 does not hold.
For sites of finite type Corollary 1.35 has the following important generalization which is the basis for all kinds of convergence theorems for spectral sequences build out of towers of local fibrations on such sites.
Proposition 1.40. — Let T be a site of finite type and U be an object ofT of cohomological dimension less than or equal to d ^ 2. Let further ^ be afibrant simplicial sheaf on T which has no nontrivial homotopy groups in dimension ^ d i.e. such that the sheaf ^^ is weakly equivalent to the point. Then no(J^(U))=pt.
Proof. — Let GP^W -^ GPW be a tower of fibrations weakly equivalent to the tower of local fibrations P^,^ -> P^^\ Since T is a site of finite type one has ^(U) ^ lim.^ GP^(U). By Corollary 1.35 and Proposition 1.36 the fibers F, of the maps GP^JTQJ) -^ GP^(U) satisfy the condition Ko(Fi)=pt for i ^ d. Therefore 7io(lim^oGP(^^(U))=7Co(GP^^(U)) and the latter set is pt by our condition on ^T.
Corollary 1.41. — For any T and U as in Proposition 1.40 and any simplicial sheaf'JT one has:
1. the map 7lo(^W(U)) -> 7Co(£<P^)(U)) is an epimorphism for i ^ d- 1 and an isomorphism for i ^ d;
2. for any x G ^T(U) the map 7C^(^T)(U), x) -^ 7^(P^)(U), x) is an epimorphism for i — k ^ d — 1 and an isomorphism for i — k ^ d.
Functoriality
We first recall briefly the standard definitions related to functoriality of sites.
Let /-1 : T2 —> TI be a functor between the underlying categories of sites Ti, T2. Associated to any such functor we have a pair of adjoint functors between the corresponding categories of presheaves of sets
^ : PreShv(T,) ^ PreShv(T,}
/, : PreShv(T^ -^ PreShv(T^)
(where f^ is just the functor given by the composition with/"1).
Definition 1.42. — A continuous map ofsites f: Ti —> T^ is a functor /-1 : T2 —>• Ti such that for any sheaf F on Ti the presheaff^F) is a sheaf on T^.
If/is a continuous map of sites, the functor/: Shv{T^) —^ Shv(T<^) has a left adjoint /*: Shv(T^) —» Shv{Tt) given by the composition of the inclusion Shv(T^) C PreShv(T^) with
the functor/^ and the functor associated sheaf a: PreShv{T^) —^ Shv(T\).
Definition 1.43. — A continuous map of sites f: Ti —> T<^ is called a morphism of sites if the functor f' : Shv{T^) —> 6%y(Ti) commutes with finite limits.
Remark 1.44. — If the topology on T^ is defined by a pretopology ([13, II.
Definition 1.3]) and the functor/"1 commutes with fiber products then/"1 defines a continuous map of sites if and only if it takes coverings (of the pretopology on Tg) to coverings (cf[l3, III. Proposition 1.6]). See [13, III. Exemple 1.9.3] for an example of a functor/-1 which takes coverings to coverings and which is not continuous.
Remark 1.45. — If the category T2 has fiber products and any representable presheaf on Ti is a sheaf then a continuous map/is a morphism of sites if and only if the functor/-1 commutes with fiber products. A more general statement can be found in (^[13, IV4.9.2]).
Example 1.46. — A typical example of a continuous map which is not a morphism of sites is given by the inclusion functor Sm/S —^ Sch/S from the category of smooth S-schemes of finite type to the category of all schemes of finite type over some base scheme S considered with Zariski (or etale, flat, Nisnevich etc.) topology ((/1.19 below).
Let/: Ti —> T2 be a continuous map of sites. Then we have a pair of adjoint functors
/* : A^Shv^) -^ A°^(Ti) / : A°^^(Ti) -^ ^Shv{T^)
between the corresponding categories of simplicial sheaves. In general neither one of them preserves weak equivalences.
Choose a resolution functor Ex for T (see 1.6). The functor/ oEx : A^kSte(Ti) —>
^Shv^T^) does preserve weak equivalences because for any weak equivalence / the morphism Ex{f) is a simplicial homotopy equivalence (cf 1.10) and the functor/
clearly preserves simplicial homotopies. Let us denote by R/:^(Ti)-^^(T2)
the functor induced by the functor f^oEx. One can easily see that R/, is the total right derived of/ in the sense of [26, 1.4]; in particular it doesn't depend on the choice of the resolution functor Ex.
The following simple result describes the basic functoriality of the simplicial homotopy categories for morphisms of sites.
Proposition 1.47. — Let f\ Ti —> T^ be a morphism of sites. Then the junctor f* preserves weak equivalences and the corresponding junctor between homotopy categories is left adjoint to R/,. If Ti —> T2 —> Tg is a composable pair of morphisms of sites then the canonical morphism of junctorsf 8.
R(?°/)*^R^°R/*
is an isomorphism.
For a site T denote by T' the site with the same underlying category considered with the trivial topology and let n: T —> T' be the canonical morphism of sites. Then n^ is the inclusion of sheaves to presheaves, TC* is the functor of associated sheaf and we have the following refinement of Proposition 1.47.
Lemma 1.48. — In the notations given above the junctor TC* :^(T')-^^(T)
is a localisation, the junctor RTT^ : ^\ (T) —> ^, (T7) is a full embedding and there is a canonical isomorphism 7C*R7^ ^ Id.
Iff is not a morphism of sites it is not clear in general whether or not R/,, has a left adjoint. There are also examples of composable pairs of continuous maps f and g such that the natural morphism R^ of)^ —> Rg^ o R/^ is not an isomorphism. We are going to define now a class of continuous maps called reasonable for which a left adjoint to R/^ always exists and the composition morphisms are isomorphisms.
Recall that for simplicial sheaves J ^ , ^ ' , the simplicial function object S(J%", j y ' ) is the simplicial set of the form
S(^T, ^\ =Hom^h^{^ x A", ^'\
Definition 1.49. — Let Ti —> T^ be a continuous map of sites. A simplicial sheaf ^ on T2 is said to be f-admissible if for any jibrant simplicial sheaf ^ on T\ and any simplicial set K the map :
n(^ x KJW) -^ Hom^^(^ x KJW) is bijective.
We say that Ts has enough f-admissibks if there is a functor adf: A^Shv^T^) —> ^Shv^T^) and a natural transformation adf —>• Id such that adf takes values in the full subcategory of objects admissible with respect to f and for any ^ on Tg the morphism adj(^) —> ^ is a weak equivalence. We then say that the pair (adf, adf —> Id) is an f-admissible resolution.
Remark 1.50. — Observe that a simplicial sheaf ^ on Ts is ^admissible if and only if for any fibrant simplicial sheaf ^ on Ti and for any weak equivalence f^) -^ ^1 with S F ' fibrant the induced map of simplicial sets S(^,/,(^)) ->
S(^, ^') is a weak equivalence.
The following two results follow immediately from the definitions (and the formal fact that for any simplicial sheaves S^ on Ti, ^ on Ts, the map n(^f xK,^(^)) —>
</W) x K, JT) is bijective).
Proposition 1.51. — Z^ Ti —> T^ be a continuous map of sites such thatT2 has enough admissibks with respect to f and (adf, adf —f Id) be an f-admissible resolution. Then the junctor /* o adf preserves weak equivalences and the induced junctor L/* : S^^Y^) —> J%^(Ti) is left
adjoint to R/^ (in particular this induced functor is independent of the/-admissible resolution).
Proposition 1.52. — Let Ti —> T^ be a continuous map of sites such that T2 has enough f-admissibks. Then a simplicial sheaf SK" on T2 is f-admissible if and only if the canonical
morphismf^^adf^)} —^/*(^) is a weak equivalence.
Lemma 1.53. — Let Ti —» T2 be a continuous map of sites and Ay be the class of f-admissibk simplicial sheaves on Tg. Then one has:
1. Ay is closed under sums;
2. for any diagram of the form ^o ^ ^i -^ - "r>l ^n —> - such that ^n € A/ and
all the morphisms u^f*{Un) are monomorphisms one has colwin^n ^ A/^*
3. for any cocartesian square of the form
^o —— ^i
i i
^2 ——— ^3
such that ^^, ^^ ^2 G Ay and both u and f^(u) are monomorphisms one has ^3 G Ay.
Proof. — The first statement is obvious. The second follows from the fact that an inverse limit of a tower of weak equivalences of simplicial sets is a weak equivalence at least if all the morphisms in the towers are fibrations.
To prove the third one, one notes that for any fibrant ^ on Ti we have a morphism of Cartesian squares of simplicial sets consisting of S(^,^(J%T)) and S(^,&(/^(^r))) respectively such that three out of four morphisms are weak equivalences and all we have to show is that the fourth one is also a weak equivalence.
This follows immediately from the fact that the maps
s(^i,./W)^S(^/*W)
s(^i, £</*W)) ^ s(^o, ^</*W))
induced by u are fibrations - the first one since/*(^) is a monomorphism and JT is fibrant and the second since u is a monomorphism and Ex{f^(J^)) is fibrant.
Proposition 1.54. — Let ^ be a simplidal sheaf on T2 such that all its terms ^ are f-admissibk. Then so is ^ .
Proof. — Let JT be a fibrant simplicial sheaf on Tp We have to show that the morphism of simplicial sets S(^< ,/*(^)) —^ S(^, ^/) is a weak equivalence for any weak equivalence ^ —> ^ ' with JT' fibrant. This morphism can be obtained by applying the total space functor to the morphism of the corresponding cosimplicial simplicial sets {cf [3, X.3]) which is a weak equivalence in the sense of [3] by the conditions of the proposition.
For any simplicial sheaf ^ and any fibrant simplicial sheaf S^ the cosimplicial simplicial set S(^,<^) is fibrant (in the sense of [3, X]). Since S(^,/,(^))=
S( /*(^)? ^) we conclude that both cosimplicial simplicial sets we consider are fibrant and our result follows now from [3, X.5.2].
Definition 1.55. — A continuous map Ti —> Tg is called reasonable if any representabk sheaf on r!^ is f-admissibk.
Example 1.56. — One may get an "unreasonable" map of sites as follows. Let /: Ti —> TS be any continuous map which is not a morphism of sites. Consider Shv{Ti) and Shv(T^) as sites with the canonical topologies. Then the functor of inverse image Shv(T^) —» Shv(T\) is an unreasonable continuous map. Note that this example also confirms that the notion of a reasonable map actually depends on sites and not just on the corresponding topoi.
Let/: Ti —> TS be a reasonable continuous map of sites. By Lemma 1.16 applied to the set y of representable sheaves there exists a functor d^ : ^Shv^P) —> ^Shv^T) and a natural transformation e^ —^ Id such that for any S^ and any n ^ 0 the sheaf of sets C^G^)^ is a direct sum of representable sheaves and the morphism
$T2(^) ~^ ^ is a trivial local fibration. Proposition 1.54 then implies that T2 has enough ^admissibles. We may sum up the situation as follows using Propositions 1.51,
1.52 and keeping previous notations.
Proposition 1.57. — Letf: Ti —>' T^ be a reasonable continuous map of sites:
1. the Junctor /* o 0^ : ^Shv^^) —^ A°^(Ti) preserves weak equivalences and the induced junctor L/* : ^(Ts) -> ^(Ti) is left adjoint to R/,;
2. if^ is a simplicial sheaf such that any term ^ of ^ is a direct sum of representabk sheaves then the canonical morphism f*(^{F)) —>/*(F) is a weak equivalence;
3. tfT\ "-^ T2 —^ TS is a composabk pair of reasonable continuous maps of sites then there are canonical isomorphisms
L(?°/)*=L/*oL^*
R(?°/)*=R?*°R/*
of functors between the corresponding homotopy categories.
Remark 1.58. — An example of a reasonable continuous map/: Ti —> T2 and a simplicial sheaf ^ on Ts such the morphism L/*(^) —»/*(^) is not a weak equivalence is given in 1.22.
Godement resolutions
The main result of this section is Theorem 1.66 below which asserts that for any site of finite type there exists a resolution functor on the category of simplicial sheaves which commutes with finite limits and takes local fibrations to global fibrations. We do not know whether the finite type assumption is really necessary for this result or not.
For any set of points cS? of T define a functor ^^ from sheaves on T to cosimplicial sheaves on T as follows. Let S be the product of 3S copies of the category of sets. A point of T is a morphism of sites Sets —> T and a set of points =S?
defines a morphism of sites p : S —> T. The corresponding adjoint pair of functors p*
and p^ gives in a standard way a cosimplicial functor with terms of the form Q^*)^1
which we denote by ^^. In most places below we omit cS? from our notations.
Proposition 1.59. — For any local fibration of locally fibrant simplicial sheaves f: J^ —> ^ the morphism
holim^[f): holim^jy -^ holim^^
is a fibration.
Proof. — By definition of local fibration the functor j&* takes local fibrations to fibrations in ^Shv^}. Since direct images preserve fibrations the composition p^
takes local fibrations to fibrations and in particular locally fibrant sheaves to fibrant sheaves. The statement of the proposition follows now from Proposition 1.22.
Proposition 1.60. — The junctor ^ ^-> holim^9^) takes weak equivalences of locally fibrant simplicial sheaves to weak equivalences of simplicial sheaves.
Proof. — One can easily see that the functors (p^}^ take weak equivalences to pointwise weak equivalences. The statement of the proposition follows now from the fact that holim preserves pointwise weak equivalences between pointwise fibrant sheaves by its definition and the corresponding result for simplicial sets (see [3, XI.5.6]).
Proposition 1.61. — Let ^9 be a cosimplicial simplicial sheaf such that all of its simplicial terms are locally jibrant and there exists s ^ 0 such that the canonical morphisms
^ -_, p^)^
are weak equivalences for all n ^ 0. Then for any point xofT the canonical morphism x\holim^} -^ holim^^9
is a weak equivalence.
Proof. — Let E^ be a resolution functor on the category of cosimplicial simplicial sets (with respect to the standard closed model structure described in [3]). Below we use the equality sign instead of specifying explicit weak equivalences. Unless otherwise specified functors on cosimplicial simplicial sets are extended to functors on cosimplicial simplicial presheaves pointwise. The functor of associated sheaf is denoted by a. We have
x^holim^9) = xa(holim^9) = x " a(holim^E^(^9))
since the functor x*a takes pointwise weak equivalences of simplicial presheaves to weak equivalences of simplicial sets. We have
x'a(holim^E^(^^) = xa(To1{E^(^9}))
since the homotopy limit is weakly equivalent to Tot for fibrant cosimplicial simplicial sets. By Lemma 1.62 we have
x'a(Tol{E^(^9))) = x^a(Tot^{E^(^9})).
Since Tot^\ involves only finite limits and functors x* and a commute with such limits we have
x^a(Tot^{E^{J^)))= Tot^a{E^{^9))).
The functor x " a commutes with finite limits and takes pointwise fibrations of simplicial presheaves to Kan fibrations of simplicial sets. In addition x^a commutes with pointwise P^. Therefore cosimplicial simplicial set x*a{E^{J^9)) satisfies the condition of Lemma 1.62 and we have
Tot^(x'a(E^(^9))) = Tol(x'a(E^{^))) = holim^xa(E^(J^))).
Finally x*a takes pointwise weak equivalences to weak equivalences and therefore holim^a(E^(^^ = holim^a^9 = holim^^\
Lemma 1.62. — Let K* be a fibrant cosimplicial simplicial set and s ^ 0 be an integer such that for any n ^ 0 the map of simplicial sets K" —> P^K^ is a weak equivalence. Then the canonical map
TotK9 -^ Tot^K9
is a weak equivalence of simplicial sets.
Proof. — Let cosks+\K^ be the cosimplicial simplicial set obtained from K* by applying the coskeleton functor to each simplicial term. Under our assumptions on K*
the canonical morphism K* —> cosks+\K.9 is a weak equivalence of cosimplicial simplicial sets. In addition, the cosimplicial simplicial set cosks+\K* is fibrant i.e. all the maps cosks+iK^1 —^ M^o^+iK* are fibrations (see [3]). To prove this fact observe that the functor cosks+\ commutes with finite limits which implies that Mncosks+\K* = cosks+\Mn¥^.
Although the coskeleton functor does not in general take Kan fibrations to Kan fibrations the following simple result holds.
Lemma 1.63. — Letf'. E —> B be a Kan jibration of Kan simplicial sets and s be an integer such that for any point x in B one has 7Ty+i(B_, x) =0. Then cosks+\{f} is again a Kan jibration.
Under our assumptions on K* we have ns+\(MnK9, x) = 0 for any point x in M^K*. This can be shown by induction on n using the intermediate objects M^K*
as in [3, Lemma 5.3, p. 278]. Therefore the maps cosks+iK^1 —> cosks+\MnK9 are fibrations and cosks+\JK' is fibrant.
For any cosimplicial simplicial set K* the canonical map 7ot(cosks+\K9) —>
Tots+\{cosks+\K*) is an isomorphism of cosimplicial simplicial sets. Since both functors Tot and Tots+\ preserve weak equivalences between fibrant objects we conclude that
Tot{K9) ^ Tot{cosk^K9)=Tot^{cosk^K9) ^ Tot^{K9).
Lemma 1.64. — For any simplicial sheaf J%T the composition p^ ->p\holim^^) -^ holim^(^^) is a weak equivalence of simplicial sheaves on S.
Proof. — This is a particular case of [23, Cor. 3.5]. In the notations of that paper one takes V=Id, F=j&* and T=j^j&*.
Recall that a set cS? of points of T is called conservative if any morphism /: F —> G of sheaves on T for which all the maps of sets x"{f) : x*F —> x^G are
isomorphisms is an isomorphism.
Proposition 1.65. — Let T be a site of finite type and 5§ be a conservative set of points of T. Then for any locally jibrant simplicial sheaf JT the canonical morphism gjy : jy —> holim^*(^) is a weak equivalence.
Proof. — We will prove this fact in several steps.
1. For any s the canonical morphism ^J^ —> holim^^jy is a weak equivalence.
Proof. — Since SS is a conservative set of points it is sufficient to show that the morphism
^(P^jT) ^p\holim^^^}
is a weak equivalence. This follows from Proposition 1.61 and Lemma 1.64.
2. The canonical morphism
holim^9^ -> holim^holim^^^) is a weak equivalence.
Proof. — By Proposition 1.59 all the simplicial sheaves hoUm^^S^ are fibrant and the morphisms between them are fibrations. Thus by [3, XI.4.1] the right hand side is pointwise weakly equivalent to Vm^holim^9^^}. We further have
lm{holim^ g^P^) = holim^ lim^P^)
^0 s^O
since holim commutes with limits. On the other hand for any n we have lim^^^jr) = (^^(limP^) = (p.p^^)
s^O s^O
since the towers of sheaves of sets (P^^), stabilize after finitely many steps for each i which implies that
lim^P^) ^ ^^.
s^O
3. By step 1 holim^^J^ is weakly equivalent to P^J^ and since it is fibrant (by Proposition 1.59) it is pointwise weakly equivalent to JE^P^^T) for any resolution functor Ex on A^67w(T). Since holim^o preserves pointwise weak equivalences between pointwise fibrant objects step 2 implies that holim^9^ is weakly equivalent to holim^oEx^J^) which is weakly equivalent to ^T by definition of site of finite type.
Theorem 1.66. — Let T be a site of finite type. Then there exists a junctor Ex^ '. A°^(T) -^ A^CT)
and a natural transformation Id —> Ex^ with the following properties:
1. Ex^ commutes with finite limits and in particular takes the final object to the final object;
2. Ex^ takes any simplicial sheaf to afibrant simplicial sheaf;
3. Ex takes local fibrations to fibrations;
4. for any JT the canonical morphism ^ —> Ex^ (J^T) is a weak equivalence.
Proof. — For a simplicial sheaf JT denote by E^J^ the simplicial sheaf associated to the simplicial presheaf of the form U i—^ Ex°°(Jy(U)) where Ex00 is a resolution functor on the category of simplicial sets satisfying the conditions of Lemma 1.67 below (note that when the topology on T can be defined by a pretopology whose covering families are all finite U H^ £^°°(^(U)) is already a simplicial sheaf since Ex00 commutes with finite limits). Let S be a conservative set of points of T.
We set
Ex^ W = holim^^E^J^).
The properties (1)-(4) for this functor follow immediately from Propositions 1.59, 1.65 and the fact that all the functors involved in the construction of Ex^ commute with finite limits.
Lemma 1.67. — There exists a functor Ex00 : A^Sets —^ ^Sets and a natural transformation Id —> Ex00 such that the following conditions hold:
1. Ex00 commutes with finite limits and in particular takes the final object to the final object;
2. Ex00 takes Kan fibrations to Kan fibrations;
3. for any simplicial set X the map X —^ Ex°°X. is a monomorphism and a weak equivalence and £y°°X is a Kan simplicial set.
Proof. — A purely combinatorial construction of Ex00 as a filtered colimit of func- tors right adjoint to certain subdivision functors can be found in [11, pp. 212-215].
2.2. A localisation theorem/or simplicial sheaves
Basic definitions and main results
Let T be a small site and let A be a set of morphisms in J^y (T). Let us recall the standard notions ofA-local objects and A-weak equivalences ((/[10] and [4, §7]).
Definition 2.1. — An object ^ of^,{T) is called A-local if for any ^ in ^(T) and, any f\ S&^ —^ cS^ in A the map
Hom^^(^ x &^ JT) -^ Hom^^ x ^ JT) is a bijection.
We write ^^(T) for the full subcategory ofA-local objects in ^(T).
Definition 2.2. — ^ morphismf: ^ -^ ^3 m A°^Shv(T) is called an A-weak equivalence if for any A-local object ^ the map
Hom^^{^ ^) -^ Hom^^^ ^) induced by f is a bijection.
Denote the class of A-weak equivalences by WA and define the class of A- fibrations FA as the class of morphisms with the right lifting property with respect to C n WA. Observe that for any ^ and any/: ^ —> ^ in A the map
^ X ^ i -^^ X c ^ 2
is an A-weak equivalence by definition.
Remark 2.3. — An object S is A-local if and only if for any A-weak equivalence /: ^T -^ ^ the induced map Hom^^^f , ^) -^ Hom^^(^, ^) is bijective.
Remark 2.4. — Let/' be the coproduct of all member of A and A' = {/'}. Then the notions of A'-local objects, A'-weak equivalences and A'-fibrations coincides with the corresponding notions associated to A. So that it is always possible to assume A has exactly one element.
The main result of this section is the following theorem.
Theorem 2.5. — For any set A the classes (WA, FA, C) define a model category structure on A°^<r). The inclusion junctor ^^(T) -^ ^(T) has a left adjoint LA which identifies
^s,?SX) wtt^ ^ localisation of^^T) with respect to A-weak equivalences.
If A consists of one element/ the functor LA will also be denoted by L/.
Remark 2.6. — This theorem appears in [5, Th. 4.6] for T the category of sets.
See also [10, §G. 2].
We also investigate the question of whether or not the A-model structure (WA, FA, C) is proper in the sense of [2, Definition 1.2]. We do not know the answer in general but we are able to prove the following result which is sufficient to demonstrate properness in the case of sites with intervals. We shall give a proof of the following result in §2.
Theorem 2.7. — For any set of morphisms A in ^(T) the closed model structure (WA, FA, C) is right proper. It is left proper if there exists a set A ofmonomorphisms in ^Shv^T) such that:
